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The established reference (see for example arXiv:0912.0651) to Taubes trick is C.H. Taubes, Seiberg Witten and Gromov invariants for symplectic 4-manifolds (2000). This book collects results from fo …

It seems your equation implies $\hat{W}(0,\mu_1)\hat{W}(0,\mu_2)=\hat{W}(0,\mu_1+\mu_2)$, which would mean that $\hat{W}(0,\mu)= e^{c\mu}$. This is not the Fourier transform of a valid marginal distri …

The factor of four and why Goldman has a different factor is explained on page 6 of Generalization of a formula of Wolpert for balanced geodesic graphs on closed hyperbolic surfaces. A derivation with …

For the record, to answer Q2, let me write down the Mehler kernel for the square root of the Fourier transform, $$K(x,y)=(2\pi)^{-1/2}\sqrt{1-i}\exp\left[{\tfrac{1}{2} i \left(x^2+y^2-2 \sqrt{2} x y\r …

V.I. Arnold's Mathematical Methods of Classical Mechanics is entirely based on the ideas and methods of symplectic geometry, such as the Birkhoff normal form, the Kolmogorov- Arnold-Moser theorem on t …

An analogue of the Huygens principle (including interference, so more precisely a Huygens-Fresnel principle) in superspace has been formulated by Gomes in arXiv:gr-qc/0602092. The formulation is used …

The gravitational or Coulomb potential has a "hidden" symmetry (hidden in the sense that it does not follow from the rotational symmetry). The resulting integral of the motion (the Runge-Lenz vector) …

To build intuition for the Planck constant $\hbar$, which I understand is the purpose of the OP, I would start by noting that $\hbar$ is not a dimensionless number: it has dimensions of energy $\times …

The symplectic counterpart of the fact that a family of commuting diagonalizable matrices is simultaneously diagonalizable is discussed in section 3.1 of On the diagonalizability of a matrix by a symp …

the $a_i$'s are complex coefficients in the asymptotic expansion $a(h)=\sum_{i=0}^\infty a_i h^i$; a more explicit expression is given on page 275-276 of Guillemin and Sternberg:

A recent (2017) overview of the status of Arnold's conjecture is given in The number of Hamiltonian fixed points on symplectically aspherical manifolds.

The master thesis Nonholonomic Dynamical Systems by Brett Ryland contains several examples of non-Hamiltonian systems from classical physics: the dynamics of a laser and the evolution of a gas flame ( …

This echos the 2017 comments, but since the question has now been bumped to the front page it might be helpful to give the actual source in Maslov's book [1]. [1] V.P. Maslov, Perturbation Theory …

The inversion is conveniently described in terms of the Fourier transform $$g(x,p)=\int dy\,e^{-iyp}G(x+y/2,x-y/2).$$ Then the composition $f(x,p)=g(x,p)\star h(x,p)$ is a matrix multiplication [1], $ …

An integrable hierarchy is another name for a system of commuting Hamiltonian flows. The word "hierarchy" is used because a countably infinite number of commuting flows is obtained recursively. [For t …